For \(n =0,1,2,\ldots ,\)  let \({\mathcal {V}}_n({\mathbb {V}}_0^{d+1}, \varpi _{{\beta },{\gamma }})\)  be the space of orthogonal polynomials of
degree \(n\)  with respect to the inner product \( {\langle }\cdot , \cdot {\rangle }_{{\beta },{\gamma }}\)  on the surface \({\mathbb {V}}_0^{d+1}\) .
If \(Q \in {\mathcal {V}}_n({\mathbb {V}}_0^{d+1}, \varpi _{{\beta },{\gamma }})\) , then \(Q(x,t)\)  is a polynomial of total degree \(n\)  in \((x,t)\) 
variables restricted on the surface of paraboloid. Since \({\mathbb {V}}_0^{d+1}\)  is a quadratic surface,
the dimension of \({\mathcal {V}}_n({\mathbb {V}}_0^{d+1}, \varpi _{{\beta },{\gamma }})\)  is equal to that of \({\mathcal {H}}_n^{d+1}\)  as
established in [1]}; that is,
\(\dim {\mathcal {V}}_n({\mathbb {V}}_0^{d+1}, \varpi _{{\beta },{\gamma }}) = \binom{n+d}{n} - \binom{n+d-2}{n-2}.\) 
